Bosonic code 

Also known as Continuous-variable (CV) quantum code, Oscillator code.
Root code for the Bosonic Kingdom

Description

Encodes logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space that contains at least one oscillator (a.k.a. bosonic mode or qumode). States of a single oscillator are elements of the Hilbert space of \(\ell^2\)-normalizable functions on \(\mathbb{R}\). Ideal codewords may not be normalizable because the space is infinite-dimensional, so approximate versions have to be constructed in practice.

Protection

Displacement error basis

An error set relevant to bosonic stabilizer codes is the set of displacement operators, a bosonic analogue of the Pauli string basis for qubit codes.

Displacement operators: For a single mode, its elements are products of exponentials of the mode's position and momentum operators, acting on the mode's position states \(|y\rangle\) for \(y\in\mathbb{R}\) as \begin{align} e^{-iq\hat{p}}\left|y\right\rangle =\left|y+q\right\rangle \,\,\text{ and }\,\,e^{iq\hat{x}}\left|y\right\rangle =e^{iq y}\left|y\right\rangle ~, \tag*{(1)}\end{align} where \(q\in\mathbb{R}\). The former is also called a translation, while the latter is called a modulation in signal processing. For multiple modes, error set elements are tensor products of elements of the single-oscillator error set, characterized by the vector of coefficients \(\xi\in\mathbb{R}^{2n}\).

The displacement error set is a unitary basis for bounded operators on the \(n\)-mode Hilbert space that is Dirac-orthonormal under the Hilbert-Schmidt inner product. Expanding a bounded operator in terms of displacements is called the Fourier-Weyl transform or Fourier-Weyl relation [2][1; Eq. (4.11)] (e.g., for the expansion of Gaussian unitary operations, see [3; Eq. (7.62)]).

There are two definitions of code distance associated with displacements. The definition inherited from qubit codes is the minimum weight of a displacement operator (i.e., number of nonzero entries in \(\xi\)) that implements a nontrivial logical operation in the code. The second definition is the minimum Euclidean distance (i.e., \(\ell^2\)-norm of \(\xi\)) such that the corresponding displacement implements a nontrivial logical operation in the code.

Loss and gain operators

An error set relevant to Fock-state bosonic codes is the set of loss operators associated with the amplitude damping (a.k.a. photon loss or attenuation) noise channel, a common form of physical noise in bosonic systems. For a single mode, loss operators are proportional to powers of the mode's annihilation operator \(a=(\hat{x}+i\hat{p})/\sqrt{2}\), where \(\hat x\) (\(\hat p\)) is the mode's position (momentum) operator, and with the power signifying the number of particles lost during the error. For multiple modes, error set elements are tensor products of elements of the single-mode error set. A definition of distance associated with this error set is the minimum weight of a loss error that implements a nontrivial logical operation in the code.

Number-phase operators

An related error set is the set of powers of the Susskind–Glogower phase operator \(\frac{1}{\sqrt{a a^\dagger}} a\) and its adjoint [46] along with Fock-space rotations generated by the occupation number operator \(a^\dagger a\). These can also be obtained from qudit Pauli matrices through a limiting procedure [6] and allow one to expand trace-class operators despite not forming an orthonormal set [7]. These operators are correspong to the number-phase interpretation, a polar-like decomposition of a single mode, complementing the cartesian-like decomposition in terms of position and momentum displacements.

Rate

The quantum capacity of the pure-loss channel [8] and the dephasing noise channel [9] are both known. The capacity of the displacement noise channel, the quantum analogue of AGWN, has been bounded using GKP codes [10,11]. Exact two-way assisted capacities have been obtained for the pure-loss channels and quantum limited amplifiers in what is known as the PLOB bound [12]. These are examples of Gaussian channels, i.e., channels that map Gaussian states to Gaussian states [1319].

Gates

Analogues of qubit Pauli operations are the displacement operations, which are unitaries generated by linear combinations of positions and momenta. Analogues of Clifford-group transformations are the Gaussian unitary transformations (a.k.a. symplectic transformations or linear canonical transformations) [20,21], which are unitaries generated by quadratic polynomials in positions and momenta. Computing using Gaussian states and Gaussian unitaries only can be efficiently simulated on a classical computer [22,23]; this remains true even if superpositions of Gaussian states are considered [24,25].A cubic phase gate is required to make a universal gate set on the oscillator [26]. More generally, controllability has only been proven when the normalizable state space is restricted to Shwartz space, the space of states with bounded moments of position and momentum [27].Measurements can be performed by homodyne and generalized homodyne measurements [28].The number-phase interpretation allows for the mapping of rotor Clifford gates into the oscillator, some of which become non-unitary (e.g., conditional occupation number addition) [29].ZX calculus has been extended to bosonic codes [30].

Notes

For an introduction to continuous-variable quantum systems, see reviews [7,3135] and books [1,36,37].See video tutorial by V. V. Albert.

Parent

  • Group-based quantum code — Group quantum codes whose physical spaces are constructed using the group of the reals \(\mathbb{R}\) under addition are bosonic codes.

Children

Cousins

  • Bosonic c-q code — Bosonic c-q codes are bosonic codes designed to transmit classical information.
  • EA bosonic code — EA bosonic codes utilize additional ancillary modes in a pre-shared entangled state, but reduce to ordinary bosonic codes when said modes are interpreted as noiseless physical modes.
  • Fermion code — Bosonic (fermionic) codes are associated with bosonic (fermionic) degrees of freedom.

References

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A. Serafini, “Quantum Continuous Variables”, (2017) DOI
[2]
K. E. Cahill and R. J. Glauber, “Ordered Expansions in Boson Amplitude Operators”, Physical Review 177, 1857 (1969) DOI
[3]
M. de Gosson, Symplectic Geometry and Quantum Mechanics (Birkhäuser Basel, 2006) DOI
[4]
L. Susskind and J. Glogower, “Quantum mechanical phase and time operator”, Physics Physique физика 1, 49 (1964) DOI
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J. Bergou and B.-G. Englert, “Operators of the phase. Fundamentals”, Annals of Physics 209, 479 (1991) DOI
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S. D. Bartlett, H. de Guise, and B. C. Sanders, “Quantum encodings in spin systems and harmonic oscillators”, Physical Review A 65, (2002) arXiv:quant-ph/0109066 DOI
[7]
V. V. Albert, “Bosonic coding: introduction and use cases”, (2022) arXiv:2211.05714
[8]
M. M. Wolf, D. Pérez-García, and G. Giedke, “Quantum Capacities of Bosonic Channels”, Physical Review Letters 98, (2007) arXiv:quant-ph/0606132 DOI
[9]
L. Lami and M. M. Wilde, “Exact solution for the quantum and private capacities of bosonic dephasing channels”, Nature Photonics 17, 525 (2023) arXiv:2205.05736 DOI
[10]
J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001) arXiv:quant-ph/0105058 DOI
[11]
K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019) arXiv:1801.07271 DOI
[12]
S. Pirandola et al., “Fundamental limits of repeaterless quantum communications”, Nature Communications 8, (2017) arXiv:1510.08863 DOI
[13]
B. Demoen, P. Vanheuverzwijn, and A. Verbeure, “Completely positive quasi-free maps of the CCR-algebra”, Reports on Mathematical Physics 15, 27 (1979) DOI
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J. Eisert and M. M. Wolf, “Gaussian quantum channels”, (2005) arXiv:quant-ph/0505151
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M. M. Wolf, “Not-So-Normal Mode Decomposition”, Physical Review Letters 100, (2008) arXiv:0707.0604 DOI
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F. Caruso et al., “Multi-mode bosonic Gaussian channels”, New Journal of Physics 10, 083030 (2008) arXiv:0804.0511 DOI
[17]
A. S. Holevo, “The Choi–Jamiolkowski forms of quantum Gaussian channels”, Journal of Mathematical Physics 52, (2011) arXiv:1004.0196 DOI
[18]
F. Caruso et al., “Optimal unitary dilation for bosonic Gaussian channels”, Physical Review A 84, (2011) arXiv:1009.1108 DOI
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J. S. Ivan, K. K. Sabapathy, and R. Simon, “Operator-sum representation for bosonic Gaussian channels”, Physical Review A 84, (2011) arXiv:1012.4266 DOI
[20]
M. Moshinsky and C. Quesne, “Linear Canonical Transformations and Their Unitary Representations”, Journal of Mathematical Physics 12, 1772 (1971) DOI
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C. Weedbrook et al., “Gaussian quantum information”, Reviews of Modern Physics 84, 621 (2012) arXiv:1110.3234 DOI
[22]
V. Veitch et al., “Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation”, New Journal of Physics 15, 013037 (2013) arXiv:1210.1783 DOI
[23]
A. Mari and J. Eisert, “Positive Wigner Functions Render Classical Simulation of Quantum Computation Efficient”, Physical Review Letters 109, (2012) arXiv:1208.3660 DOI
[24]
U. Chabaud et al., “Classical simulation of Gaussian quantum circuits with non-Gaussian input states”, Physical Review Research 3, (2021) arXiv:2010.14363 DOI
[25]
B. Dias and R. Koenig, “Classical simulation of non-Gaussian bosonic circuits”, (2024) arXiv:2403.19059
[26]
S. L. Braunstein and P. van Loock, “Quantum information with continuous variables”, Reviews of Modern Physics 77, 513 (2005) arXiv:quant-ph/0410100 DOI
[27]
R.-B. Wu, T.-J. Tarn, and C.-W. Li, “Smooth controllability of infinite-dimensional quantum-mechanical systems”, Physical Review A 73, (2006) arXiv:quant-ph/0505063 DOI
[28]
A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography”, Reviews of Modern Physics 81, 299 (2009) arXiv:quant-ph/0511044 DOI
[29]
Y. Xu, Y. Wang, and V. V. Albert, “Clifford operations and homological codes for rotors and oscillators”, (2024) arXiv:2311.07679
[30]
H. Nagayoshi et al., “ZX Graphical Calculus for Continuous-Variable Quantum Processes”, (2024) arXiv:2405.07246
[31]
B. M. Terhal, J. Conrad, and C. Vuillot, “Towards scalable bosonic quantum error correction”, Quantum Science and Technology 5, 043001 (2020) arXiv:2002.11008 DOI
[32]
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W. Cai et al., “Bosonic quantum error correction codes in superconducting quantum circuits”, Fundamental Research 1, 50 (2021) arXiv:2010.08699 DOI
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K. Noh, “Quantum Computation and Communication in Bosonic Systems”, (2021) arXiv:2103.09445
[35]
S. M. Girvin, “Introduction to quantum error correction and fault tolerance”, SciPost Physics Lecture Notes (2023) arXiv:2111.08894 DOI
[36]
N. J. Cerf, G. Leuchs, and E. S. Polzik, Quantum Information with Continuous Variables of Atoms and Light (PUBLISHED BY IMPERIAL COLLEGE PRESS AND DISTRIBUTED BY WORLD SCIENTIFIC PUBLISHING CO., 2007) DOI
[37]
A. F. Kockum et al., “Lecture notes on quantum computing”, (2024) arXiv:2311.08445
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Zoo Code ID: oscillators

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“Bosonic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/oscillators
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@incollection{eczoo_oscillators, title={Bosonic code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/oscillators} }
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